Nivolumab, Carboplatin, and Paclitaxel in Treating Patients With Stage III-IV Head and Neck Squamous Cell Carcinoma That Can Be Removed by Surgery

Inclusion Criteria: * Patients must be 18 years of age and older. * Pathologically confirmed SCCHN, not previously treated, with a plan to undergo surgery * Patients who have stage III-IV disease without distant metastases (M0) of 1) oral cavity, 2) larynx, 3) hypopharynx 4) oropharynx (human papillomavirus \[HPV\] neg) using American Joint Committee on Cancer (AJCC) 8th edition * Patients who have oropharyngeal cancer that HPV positive, stage II-III disease without distant metastases (M0) using AJCC 8th edition * All patients with oropharyngeal SCCHN must be tested for HPV (by p16 and/or HPV in situ hybridization \[ISH\] or polymerase chain reaction \[PCR\]) * Patients must be evaluated by a head and neck surgeon and be deemed surgically resectable at baseline * Tumor sample must be available for HPV p16 and PD-L1 testing and if oropharyngeal, must be tested for HPV p16 * Eastern Cooperative Oncology Group (ECOG) performance status 0-1 * While blood cells 2000/ul or more * Absolute neutrophil count 1500/ul or more * Platelets 100,000/ul or more * Hemoglobin 9 g/dl or more; (transfusion permitted) * Bilirubin less than or equal to 1.5 x the upper limit of normal (except subjects with Gilbert syndrome, who can have total bilirubin \< 3 mg/dl) * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) less than or equal to 3 x the upper limit of normal * Glomerular filtration rate (GFR) greater than or equal to 40 ml/min using the Cockcroft-Gault formula or serum cr…